annotate bk/bk.org @ 6:b2f55bcf6853

fixed comments
author Robert McIntyre <rlm@mit.edu>
date Fri, 28 Oct 2011 04:56:15 -0700
parents 44d3dc936f6a
children
rev   line source
rlm@0 1 #+TITLE: Bugs in Quantum Mechanics
rlm@0 2 #+AUTHOR: Dylan Holmes
rlm@0 3 #+SETUPFILE: ../../aurellem/org/setup.org
rlm@0 4 #+INCLUDE: ../../aurellem/org/level-0.org
rlm@0 5
rlm@0 6 #Bugs in the Quantum-Mechanical Momentum Operator
rlm@0 7
rlm@0 8
rlm@0 9 I studied quantum mechanics the same way I study most subjects\mdash{}
rlm@0 10 by collecting (and squashing) bugs in my understanding. One of these
rlm@0 11 bugs persisted throughout two semesters of
rlm@0 12 quantum mechanics coursework until I finally found
rlm@0 13 the paper
rlm@0 14 [[http://arxiv.org/abs/quant-ph/0103153][/Self-adjoint extensions of operators and the teaching of quantum
rlm@0 15 mechanics/]], which helped me stamp out the bug entirely. I decided to
rlm@0 16 write an article about the problem and its solution for a number of reasons:
rlm@0 17
rlm@0 18 - Although the paper was not unreasonably dense, it was written for
rlm@0 19 teachers. I wanted to write an article for students.
rlm@0 20 - I wanted to popularize the problem and its solution because other
rlm@0 21 explanations are currently too hard to find. (Even Shankar's
rlm@0 22 excellent [[http://books.google.com/books/about/Principles_of_quantum_mechanics.html?id=2zypV5EbKuIC][textbook]] doesn't mention it.)
rlm@0 23 - I wanted to check that the bug was indeed entirely
rlm@0 24 eradicated. Attempting an explanation is my way of making
rlm@0 25 sure.
rlm@0 26
rlm@0 27 * COMMENT
rlm@0 28 I recommend the
rlm@0 29 paper not only for students who are learning
rlm@0 30 quantum mechanics, but especially for teachers interested in debugging
rlm@0 31 them.
rlm@0 32
rlm@0 33 * COMMENT
rlm@0 34 On my first exam in quantum mechanics, my professor asked us to
rlm@0 35 describe how certain measurements would affect a particle in a
rlm@0 36 box. Many of these measurement questions required routine application
rlm@0 37 of skills we had recently learned\mdash{}first, you recall (or
rlm@0 38 calculate) the eigenstates of the quantity
rlm@0 39 to be measured; second, you write the given state as a linear
rlm@0 40 sum of these eigenstates\mdash{} the coefficients on each term give
rlm@0 41 the probability amplitude.
rlm@0 42
rlm@0 43 * The infinite square well potential
rlm@0 44 There is a particle in a one-dimensional potential well that has
rlm@0 45 infinitely high walls and finite width \(a\). This means that the
rlm@0 46 particle exists in a potential[fn:coords][fn:infinity]
rlm@0 47
rlm@0 48
rlm@0 49 \(V(x)=\begin{cases}0,&\text{for }\;0< x< a;\\\infty,&\text{for
rlm@0 50 }\;x<0\text{ or }x>a.\end{cases}\)
rlm@0 51
rlm@0 52 The Schr\ouml{}dinger equation describes how the particle's state
rlm@0 53 \(|\psi\rangle\) will change over time in this system.
rlm@0 54
rlm@0 55 \(\begin{eqnarray}
rlm@0 56 i\hbar \frac{\partial}{\partial t}|\psi\rangle &=&
rlm@0 57 H |\psi\rangle \equiv\frac{P^2}{2m}|\psi\rangle+V|\psi\rangle \end{eqnarray}\)
rlm@0 58
rlm@0 59 This is a differential equation whose solutions are the physically
rlm@0 60 allowed states for the particle in this system. Like any differential
rlm@0 61 equation,
rlm@0 62
rlm@0 63
rlm@0 64 Like any differential equation, the Schr\ouml{}dinger equation
rlm@0 65 #; physically allowed states are those that change in physically
rlm@0 66 #allowed ways.
rlm@0 67
rlm@0 68
rlm@0 69 ** Boundary conditions
rlm@0 70 Because the potential is infinite everywhere except within the well,
rlm@0 71 a realistic particle must be confined to exist only within the
rlm@0 72 well\mdash{}its wavefunction must be zero everywhere beyond the walls
rlm@0 73 of the well.
rlm@0 74
rlm@0 75
rlm@0 76 [fn:coords] I chose my coordinate system so that the well extends from
rlm@0 77 \(0<x<a\). Others choose a coordinate system so that the well extends from
rlm@0 78 \(-\frac{a}{2}<x<\frac{a}{2}\). Although both coordinate systems describe the same physical
rlm@0 79 situation, they give different-looking answers.
rlm@0 80
rlm@0 81 [fn:infinity] Of course, infinite potentials are not
rlm@0 82 realistic. Instead, they are useful approximations to finite
rlm@0 83 potentials when \ldquo{}infinity\rdquo{} and \ldquo{}the actual height
rlm@0 84 of the well\rdquo{} are close enough for your own practical
rlm@0 85 purposes. Having introduced a physical impossibility into the problem
rlm@0 86 already, we don't expect to get physically realistic solutions; we
rlm@0 87 just expect to get mathematically consistent ones. The forthcoming
rlm@0 88 trouble is that we don't.